7.2 Rate of Change and Slope of a Line

Definition and Formula for Rate of Change

• The rate of change represents the ratio of the change in a response variable ($y$) to the change in an explanatory variable ($x$) as they vary from $(x_1, y_1)$ to $(x_2, y_2)$.

• General Formula:     $\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}$

• Key Examples:

  • Bank Robberies (2006–2018): Declined from $6985$ to $2975$. The rate of change is approximately $-334.2\,\text{robberies/year}$, representing a yearly decline.

  • Karate Classes: Cost of $12\,\text{classes}$ is $158\,\text{dollars}$; $20\,\text{classes}$ is $230\,\text{dollars}$. The rate of change is exact at $9\,\text{dollars/class}$.

Linear Associations and Rate of Change

• Constant Rate of Change: If the rate of change between two variables is constant, there is an exact linear association between them.

• Correlation Coefficient ($r$): For a constant rate of change in a positive direction, $r = 1$.

• Explanatory vs. Response Variables: In a travel scenario where a student drives at $50\,mph$, time ($t$) is the explanatory variable and distance ($d$) is the response variable.

• Directional Correlation:

  • Positive Association: Higher values of one variable relate to higher values of the other; the rate of change is positive.

  • Negative Association: Higher values of one variable relate to lower values of the other; the rate of change is negative.

Estimating Rates from Linear Models

• Bald Eagle Nests in Wisconsin: Using estimated points $(0, 36)$ and $(400, 280)$, the rate of change is approximately $0.61\,\text{nestlings/nest}$.

• Sony Employment: Using points $(2011, 165)$ and $(2016, 129)$ (in thousands), the rate of change is $-7.2\,\text{thousand employees/year}$.

Slope of a Nonvertical Line

• The slope ($m$) is defined as the rate of change of $y$ with respect to $x$, often referred to as "rise over run":     $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

• Increasing Lines: Have a positive slope.

• Decreasing Lines: Have a negative slope.

• Steepness: The steepness of a line is determined by the absolute value of its slope ($|m|$). A larger absolute value indicates a steeper line.

Horizontal and Vertical Lines

• Horizontal Lines: Contain points with identical $y$-coordinates (e.g., $(3, 2)$ and $(7, 2)$). The slope is zero ($m = 0$).

• Vertical Lines: Contain points with identical $x$-coordinates (e.g., $(3, 1)$ and $(3, 5)$). The slope is undefined because the denominator (run) is zero.